D-brane monodromies from a matrix-factorization perspective

0.64 MB·English

by  Hans Jockers

hep-th/0612095
CERN-PH-TH/2006-256
D-brane monodromies from
a matrix-factorization perspective
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Hans Jockers

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Department of Physics, Theory Division
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v CERN, Geneva, Switzerland
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h Abstract
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p The aim of this work is to analyze K¨ahler moduli space monodromies of string
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h compactifications. This is achieved by investigating the monodromy action upon D-
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v brane probes, which we model in the Landau-Ginzburg phase in terms of matrix
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X factorizations. The two-dimensional cubic torus and the quintic Calabi-Yau hyper-
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a surface serve as our two prime examples.
December, 2006
1. Introduction
The discovery and investigation of D-branes have given as some insight into the
non-perturbative structure of string theory and have improved our understanding of
string dualities. However, despite of this success our view upon many aspects of
D-branes is still rather limited.
For instance many properties of D-branes in string compactifications are only
qualified in certain regions of the string moduli space, such as the geometric regime,
where the compactification space is taken to be large compared to the string scale and
hence string corrections are suppressed. These scenarios allow us to treat D-branes
semi-classically and to apply geometric methods. However, in other regions of the
moduli space we cannot neglect stringy quantum corrections [1,2,3], and therefore it is
necessary todescribeD-braneswiththemachineryofboundaryconformalfieldtheory.
In principal boundary conformal field theories constitute a suitable description for
generic values of the moduli. However, in practice these methods are only applicable
atspecialpointsinthemodulispace, whereduetoenhancedsymmetriestheconformal
field theory becomes rational and hence solvable [4,5,6]. Thus studying D-branes in
string compactifications for generic moduli remains a challenge.
Recently matrix factorizations have emerged as yet another tool to study D-
branes [7,8,9,10,11,12,13] . They model branes in Landau-Ginzburg theories, which
describe string compactifications on hypersurfaces in a non-geometric regime of the
K¨ahler moduli space [14]. In the context of Landau-Ginzburg models we are still
able to continuously vary both bulk complex structure moduli, realized in terms of
deformations of the Landau-Ginzburg superpotential, and D-brane moduli, encoded
in the matrix factorization [15,16,17]. Furthermore, we can even study obstructed
moduli and their associated effective superpotentials [15,18,16,19,20,21].
These Landau-Ginzburg theories are believed to flow to an infrared conformal
fixed point. Since this flow is rather complicated we use here the framework of topo-
logical Landau-Ginzburg theories, which compute quantities invariant with respect to
the renormalization group.
The goal of this work is to transport brane probes in the K¨ahler moduli space so
as to explore its global structure. But instead of considering an arbitrary path in the
moduli space [22] (cf. also refs. [1,3,23,24,25,26]), we are less ambitious and analyze
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branes as we move along a closed path with base point at the Landau-Ginzburg
phase in the K¨ahler moduli space. This corresponds to determining upon matrix
factorizations the action of monodromies induced from moduli space singularities. A
similaranalysis has been carried out in refs. [27,28,29,30],where the large radius point
is chosen as a base point and where the monodromies act upon complexes of coherent
sheaves.1 This work should be seen complementary to the large radius results as some
of the calculations are more tractable in the language of matrix factorizations.
The outline of the paper is as follows. In section 2 we mainly review matrix
factorizations in Landau-Ginzburg orbifolds in order to set our conventions and to
introduce the notation. In particular we focus on equivariant matrix factorizations
[11,33,17] and their gradings [33], as these properties play an important role in the
D-brane monodromy analysis.
Then we turn to the structure of the K¨ahler moduli space of Calabi-Yau hy-
persurfaces from a gauged linear σ-model point of view [14]. Typically one obtains
three kinds of singularities in the K¨ahler moduli space, namely the large radius, the
Landau-Ginzburg and the conifold singularity. In section 3 we investigate in detail
the monodromies of these singularities acting upon matrix factorizations.
Insection4 weemploythedeveloped techniques andstudy D-brane monodromies
on the moduli space of the cubic torus. The matrix factorizations of the cubic torus
are well-understood [16,17], and hence the torus serves as good first example to study
the effect of monodromies on matrix factorizations. We also demonstrate that the
results are compatible with the expected transformation behavior of D-brane charges.
Finally we show the connection of the K¨ahler moduli space as seen from the gauged
linear σ-model [14] to the Teichmu¨ller space of the two-dimensional torus [34].
We turn towards our second example, the quintic Calabi-Yau hypersurface, in
section 5. We explicitly address the action of the monodromies upon two types of
matrix factorizations of the quintic. Again we verify our results by comparing with
the monodromy transformations of the D-brane charges presented in ref. [1,32].
In section 6 we present our conclusions and in appendix A we have collected the
open-string cohomology elements used in section 4.
1 OnthelevelofD-branechargesmonodromieshavealsobeenstudiedinrefs.[1,25,31,32].
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2. D-branes in Landau-Ginzburg orbifold theories
In order to set the stage for the forthcoming analysis we review the notion of
B-type branes in the context of topological Landau-Ginzburg orbifolds. By now it
is well-known [8,9,11,12,13] that B-branes in Landau-Ginzburg theories are given by
matrix factorizations of the Landau-Ginzburg superpotential, W. In this section we
recapitulate the aspects which are important for this work.
2.1. Matrix factorizations and open-string states
A B-type brane, P, in the topological Landau-Ginzburg theory with homoge-
nous Landau-Ginzburg superpotential, W(x), is realized as matrix, Q , and a linear
P
involution, σ , i.e. σ2 = 1, such that [8,9,11,12,13]
P P
Q2P(x) = W(x)·12n×2n , σP QP +QP σP = 0. (2.1)
Here the 2n × 2n matrix, Q , has polynomial entries in the bulk chiral Landau-
P
Ginzburg fields, xℓ. Furthermore, two matrix factorizations, (QP,σP) and (QP′,σP′),
are gauge-equivalent, i.e. they describe the same brane, if they are related by an
invertible 2n×2n matrix, U(x),2
QP′(x) = U(x)QP(x)U−1(x) , σP′ = U(x)σP U−1(x) . (2.2)
From a given matrix factorization, (Q ,σ ), of a brane, P, we can immediately
P P
construct the matrix factorization, (Q ,σ ), of the anti-brane, P¯, by acting with the
P¯ P¯
operator, T:
T : P 7→ P¯ , (Q ,σ ) 7→ (Q ,−σ ) . (2.3)
P P P P
Thus the operator, T, generates the matrix factorization of the anti-brane.
ThephysicalstringstatesinthetopologicalLandau-Ginzburgtheoryariseasnon-
trivial cohomology elements of the BRST operator. For open-string states, Θ , of
(P,R)
strings stretching from the brane, P, to the brane, R, the BRST operator is given by
D Θ = Q Θ −σ Θ σ Q . (2.4)
(P,R) (P,R) R (P,R) R (P,R) P P
2 Invertible as a matrix in the ring of polynomials in xℓ.
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It is straight forward to check that the BRST operator, D , squares to zero.
(P,R)
Furthermore, we observe that the open-string states, Θ , split into bosonic
(P,R)
states, Φ , and fermionic states, Ψ , which differ by their eigenvalues ±1 with
(P,R) (P,R)
respect to the involutions of the attached branes:
σ Φ σ = +Φ , σ Ψ σ = −Ψ . (2.5)
R (P,R) P (P,R) R (P,R) P (P,R)
In the paper we also use an equivalent description for the matrix factorization,
(Q ,σ ), which arises as follows: Due to the fact that the matrix, Q , anti-commutes
P P P
with the involution, σ , we can always find a gauge in which the involution, σ , takes
P P
the block diagonal form σP = Diag(1n×n,−1n×n). In this gauge the matrix, QP,
decomposes into two n×n matrices according to3
0 J (x)
Q (x) = P . (2.6)
P E (x) 0
P
(cid:18) (cid:19)
Thus wecanalternativelydescribe thebrane, P, intermsofthematrixpair, (J ,E ),
P P
which then fulfills
JP(x)EP(x) = EP(x)JP(x) = W(x)·1n×n . (2.7)
In this description the operator, T, which maps branes to their anti-branes, becomes
T : P 7→ P¯ , (J ,E ) 7→ (J ,E ) = (−E ,−J ) . (2.8)
P P P¯ P¯ P P
Moreover, bosonic and fermionic open-string states, Φ = (φ ,φ ) and Ψ =
(P,R) 0 1 (P,R)
(ψ ,ψ ), decompose also into two matrices, and the open-string BRST operator,
0 1
D , reads
(P,R)
D Φ = D (φ ,φ ) = (J φ −φ J ,E φ −φ E ) ,
(P,R) (P,R) (P,R) 0 1 R 0 1 P R 1 0 P
(2.9)
D Ψ = D (ψ ,ψ ) = (E ψ +ψ J ,J ψ +ψ E ) .
(P,R) (P,R) (P,R) 0 1 R 0 1 P R 1 0 P
3 Note that the block-diagonal form of the involution, σP, corresponds to a partial gauge
fixing, which is preserved by gauge transformations (2.2) with block-diagonal matrices, U =
Diag(Vn×n,Wn×n). Here the n×n matrices, Vn×n and Wn×n, are invertible again in the
ring of polynomials in xℓ.
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2.2. R-charge assignments
For the Landau-Ginzburg model to flow to a non-trivial conformal IR fixed point,
it is necessary for the theory to have a (non-anomalous) U(1) R-symmetry. With
respect tothisU(1)symmetrythebulkLandau-Ginzburg superpotential hasR-charge
assignment +2. Hence for a homogenous superpotential, W(x), of degree d the bulk
chiral fields, x , have R-charge +2.4
ℓ d
For Landau-Ginzburg theories with branes it is also necessary to extend the U(1)
R-symmetry of the bulk to the boundary. This corresponds to requiring that we
can find a U(1) representation, ρ (θ), such that the matrix, Q , which according to
P P
eq. (2.1) has R-charge +1, transforms with respect to the U(1) R-symmetry as [33]5
ρP(θ)QP(e2idθx)ρ−P1(θ) = eiθQP(x) . (2.10)
Here the representation, ρP(0), obeys ρP(0) = 12n×2n and ρP(πd) = 12n×2n for even
d whereas ρP(2πd) = 12n×2n for odd d.
For us it is important that the representations, ρ (θ) and ρ (θ), of the branes,
P R
P and R, assign also the R-charge, q , to the open-string states, Θ ,
Θ(P,R) (P,R)
ρR(θ)Θ(P,R)(e2idθx)ρ−P1(θ) = eiθqΘ(P,R)Θ(P,R)(x) . (2.11)
2.3. Equivariant matrix factorizations
Ultimately we want to study monodromies in the K¨ahler moduli space of Calabi-
Yau compactifications. For the compactifications considered in this work the Landau-
Ginzburg phase is realized as a Landau-Ginzburg orbifold [14]. The orbifold group,
Z , acts on the bulk chiral fields, x , as
d ℓ
xℓ 7→ ωkxℓ , ω = e2dπi , k ∈ Zd . (2.12)
4 In this paper we consider only homogenous Landau-Ginzburg superpotentials. The
generalization to quasi-homogenous superpotentials is straight forward.
5 Wealwayschooseagaugeforthematrixfactorization,QP,suchthattherepresentation,
ρ(θ), is diagonal and x-independent (cf. ref. [33]).
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In this context branes are characterized by Z -equivariant matrix factorizations. This
d
means we need to add to the data of the brane, P, a Z representation, RP, such that
d
the matrix, Q , fulfills the equivariance condition [11,33,17]:
P
RP(k)Q (ωkx)RP(−k) = Q (x) . (2.13)
P P
In terms of the matrices, (J ,E ), the representations, RP, splits into two parts, RP
P P 0
and RP, and the equivariance condition (2.13) becomes
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J (x) = RP(k)J (ωkx)RP(−k) ,
P 0 P 1
(2.14)
E (x) = RP(k)E (ωkx)RP(−k) .
P 1 P 0
The expression (2.13) resembles closely the transformation behavior (2.10) of the
matrix, Q , with respect to the U(1) R-symmetry. Indeed for irreducible matrix
P
factorizations the representation, RP, are related to the U(1) representation, ρ , by
P
[33]
λ d
R(k) = eiπkλPρ(πk)σk , a = P ∈ Z . (2.15)
P 2
Here λ denotes the grade of the equivariant matrix factorization, which is constraint
P
by RP(d) = 12n×2n. Thus for each irreducible matrix, QP, there are d inequivalent
Z representations, RPa, which give rise to d different equivariant branes, P , in the
d a
orbifold theory. Given an equivariant brane, P, we simply obtain the other branes,
P , in the same equivariant orbit by
a
RPa(k) = ωakRP(k) . (2.16)
As the representations, RP, distinguishes among the branes in the equivariant
orbit we must also adjust the notion of open-string states. Therefore induced from
eq. (2.13) we impose on open-string states, Θ , the condition
(P,R)
RR(k)Θ (ωkx)RP(−k) = Θ (x) . (2.17)
(P,R) (P,R)
2.4. Gradings of branes
Finally let us discuss one additional refinement in the description of branes.
We have seen that branes are equipped with a grade, λ , which, so far, has been
P
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ambiguous up to shifts of even integers. As explained in refs. [35,36] this ambiguity is
not important as long as we analyzethe physics of a single brane but becomes relevant
for the analysis of open strings stretching between different branes. Thus in order to
keep track of this ambiguity, we assign to each brane an integer, n, and denote the
graded brane by P[n]. The grading, n, is the integer offset of the grade, λ . If we
P
perform the shift, λ → λ −1, we observe in eq. (2.15)that thisamounts to changing
P P
the sign of the involution, σ , i.e. σ → −σ . Thus according to eq. (2.3) the brane,
P P P
P[1], is the anti-brane of P[0], and hence we identify the operator, T, which maps
branes to anti-branes, with the translation operator for the integer grading, n:
T : P[n] 7→ P[n+1] . (2.18)
Note that in the following we abbreviate the branes, P[0] and P[1], by the short-hand
notation, P and P¯.
As a consequence of the interplay of the integer grading, n, and the grade, λ ,
P
we also obtain the relation
P [n] = P [n−2] . (2.19)
a+d a
Furthermore, for even degrees, d, we find that branes and anti-branes are in the same
equivariant orbit because the anti-brane, P¯a, coincides with the brane, Pa−d/2.
With these definitions at hand we can now assign integer gradings to open-string
states. Namely, the grading, p, of an open-string state, Θ , with R charge, q ,
(P,R) Θ(P,R)
arises as [33]
p = λ −λ +q . (2.20)
R P Θ
(P,R)
For odd and even integers, p, the open-string states are bosonic and fermionic respec-
tively. Thus, the integer grading, p, is compatible with the statistics of open-string
states. We denote the space of open-string states at grading, p, by Extp(P,R) and
for p = 0 by Hom(P,R) = Ext0(P,R). Due to eq. (2.20) the open-string states at
different gradings are related by
Extp(P,R) ≃ Hom(P[−p],R) ≃ Hom(P,R[p]) . (2.21)
Allthose described ingredients arecaptured ina graded category[37,38,26,39,22],
where the objects are matrix factorizations, the morphisms between objects are open-
string states, and finally the shift functor is the operator, T. For us it is important to
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note that in the category of matrix factorizations of topological B-banes, in addition
to the gauge equivalences (2.2), two matrix factorizations are also equivalent if they
only differ by blocks of trivial matrix factorizations [37,15,38]
0 1 0 W
Q = , Q = . (2.22)
W W 0 W¯ 1 0
(cid:18) (cid:19) (cid:18) (cid:19)
Physically the trivial matrix factorization, Q , corresponds to a trivial brane-anti-
W
brane pair, which annihilates to the vacuum.
3. D-brane monodromies in the K¨ahler moduli space
In this section we introduce the tools needed to study D-brane monodromies in
the K¨ahler moduli space of hypersurfaces embedded in (weighted) projective spaces.
These geometries have a Landau-Ginzburg orbifold phase [14,22], in which matrix
factorizations describe D-branes, and hence they are suitable to study D-brane mon-
odromies from a matrix factorization perspective.
3.1. The Ka¨hler moduli space and D-brane monodromies
In this paper the cubic torus in CP2 and the quintic hypersurface in CP4 serve as
our working examples, but the following discussion generalizes to many other Calabi-
Yau hypersurfaces as well.
Compactifications of both geometries depend on a single (complexified) K¨ahler
modulusandtheK¨ahlermodulispacebecomessingularatthreedistinctpoints. There
is the large radius point, where the volume of the compactification space becomes infi-
nite, then there is the conifold point, where the (quantum) volume of the hypersurface
shrinks to zero size while the (quantum) volume of the lower even dimensional cycles
stays finite [40], and finally there is the Landau-Ginzburg point, where the singular-
ity in the moduli space arises from a global discrete symmetry of the theory. The
structure of the K¨ahler moduli space is schematically depicted in Fig. 1 (a).
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(a) (b)
LR
LR
C
C
LG LG
Fig. 1: (a) The figure illustrates the complex one dimensional
K¨ahler moduli space of a Calabi-Yau hypersurface with the large
radius (LR), the Landau-Ginzburg (LG) and the conifold (C) singu-
larity. (b) Here we show the three non-trivial loops in the K¨ahler
modulispacealongwhichwetransportbraneprobes. Thebasepoint
of these loops is in the vicinity of the Landau-Ginzburg point, where
branes are given in terms of matrix factorizations.
In the topological B-model the dependence of branes on K¨ahler moduli is rather
mild. For instance a brane probe transported along a closed contractible loop is
expected to come back unchanged. If, however, the loop is non-contractible, that is
to say if we encircle one of the above mentioned singularities, then, in general, the
original brane configuration is changed. This, however, does not imply that we get a
new theory with different branes. Instead, it just means that the monodromy of the
singularity maps individual branes to other branes within the same theory [28].
Note that for physical branes there is a stronger dependence on the K¨ahler mod-
uli, as one also has to take into account the notion of Π-stability [41,27,28], i.e. a
physical brane probe can decay as it crosses a line of marginal stability in the K¨ahler
moduli space. However, we limit our analysis to topological branes and hence we do
not address this issue here.
Our next task is to discuss the D-brane monodromies arising from the different
singularities. As we focus on branes given by matrix factorizations, the base point for
the non-contractible loops is located near the Landau-Ginzburg point as depicted in
Fig. 1 (b).
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